The zipper macro and zipper moves

I continue from the post Generating set of Reidemeister moves for graphic lambda crossings , where the “crossing macros” over graphic lambda calculus were discussed.

Another interesting macro  (over graphic lambda calculus) is the zipper, together with its associated zipper moves.

Let’s take n \geq 2 a natural number and let’s consider the following graph in GRAPH, called the n-zipper:

At the left is the n-zipper graph; at the right is a NOTATION for it, or a macro.  We could as well take n =2, with obvious modifications of the figure, so the 2-zipper exists. Even n=1 makes sense, but the 1-zipper is kind of degenerate, see later.

There is a graphic beta move which we can perform on the n-zipper graph. In the following picture I figured in red the place where the graphic beta move is applied.

In terms of zipper notation this graphic beta move has the following appearance:

We see that a n-zipper transforms into a (n-1)-zipper plus an arrow. We may repeat this move, as long as we can. What is the result? A n-zipper move:

The 1-zipper move, called ZIP_{1} is just the graphic beta move, which transforms the 1-zipper into two arrows.

Nice, now what can we do with zippers and crossings?

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